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### Division by Zero: Indeterminate or Undefined?

```
Date: 02/23/2002 at 16:21:24
From: Brant Langer Gurganus
Subject: My theories on Zero

In a previous response on a related topic, you said that division is
defined in terms of reverse multiplication, and multiplication is
defined in terms of repeated addition.  Therefore, division is
defined in terms of repeated subtraction.

If so, then division is a count of how many times a number may be
subtracted from another number until the original number equals zero.

Case 1: 0/X; X is not 0
Operation:        Count:
0                 0 (it's already at 0 so 0/x=0)
0-X = -X          (past 0)

Case 2: X/0; X is not 0
Operation:        Count:
X                 0
X-0 = X           1
X-0 = X           2
.                 .
.                 .
.                 .
X-0 = X           infinity (not undefined)

Case 3: 0/0
Operation:        Count:
0                 0
0-0 = 0           1
.                 .
.                 .
.                 .
0-0 = 0           infinity (0/0 can have all numbers
as solutions)

Based on my cases above:

1. 0/x = 0

2. x/0 = infinity

3. 0/0 = one or more elements of the set of all numbers
(depending on context)

Can you comment on these conclusions?
```

```
Date: 02/23/2002 at 17:41:35
From: Doctor Rick
Subject: Re: My theories on Zero

Hi, Brant.

Your analysis seems like a reasonable way to describe and explain the
results of dividing by zero. The only problem I have is that your use
of the word "infinity" for both "infinitely large (greater than any
number)" (what we call "undefined") and "can have any number as a
solution" (what we call "indeterminate") is inconsistent and therefore
confusing. See our explanations here:

Dividing by 0 - Dr. Math FAQ
http://mathforum.org/dr.math/faq/faq.divideby0.html

The indeterminate nature of 0/0
http://mathforum.org/dr.math/problems/rob.12.21.00.html

Defining 0/0
http://mathforum.org/dr.math/problems/howard.01.29.01.html

- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 02/23/2002 at 19:04:08
From: Brant Langer Gurganus
Subject: My theories on Zero

What I meant was that since no matter how many times 0 is subtracted
from 0, it will always be 0; so (expanding beyond integer division)
all numbers can be a solution to 0/0. The "one or more solutions"
depends on the context of the problem.

Thanks for the quick reply.
```

```
Date: 02/23/2002 at 21:08:29
From: Doctor Rick
Subject: Re: My theories on Zero

Hi, Brant.

I know what you meant by "one or more solutions," and you're right. I
hope you read the archive items I listed, which clarify how this works
out in practice.

As I put it, "indeterminate" means that there may be a definite
answer to the problem you are working on, but you'll have to back up
and find another way to discover it; dividing zero by zero cannot tell
you the answer. It's sort of a "road closed" sign.

A similar statement applies to "undefined." Rather than say that it
takes an infinite number of subtractions of zero to get x down to 0,
we have to say that NO number of subtractions of zero will get x down
to 0. The answer to x/0 is not a number. That's another good way to
say "undefined," and some computer languages call it this.

- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Number Theory
Middle School Number Sense/About Numbers

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